Using Chebyshev's inequalities in problems of designing complex technical systems
Abstract
Using Chebyshev's inequalities in problems of designing complex technical systems
Incoming article date: 24.09.2024The current situation in the practice of designing complex technical systems with metrological support is characterized by the following important features: a) the initial information that can actually be collected and prepared at the early stages of design for solving probabilistic problems turns out, as a rule, to be incomplete, inaccurate and, to a high degree, uncertain; b) the form of specifying the initial information (in the form of constraints) in problems can be very diverse: average and dispersion characteristics or functions of them, measurement errors or functions of them, characteristics specified by a probability measure, etc. These circumstances necessitate the formulation and study of new mathematical problems of characterizing distribution laws and developing methods and algorithms for solving them, taking into account the constraints on the value and nature of change of the determining parameter (random variable) of a complex technical system. As a generalized integral characteristic of the determining parameter, the law of its distribution is chosen, which, as is commonly believed, fully characterizes the random variable under study. The purpose of this work is to develop a method that allows constructing distribution laws of the determining parameter of a complex technical system using the minimum amount of available information based on the application of Chebyshev inequalities. A method for characterizing the distribution law by the property of maximum entropy is presented, designed to model the determining parameter of complex technical systems with metrological support. Unlike the classical characterization method, the proposed method is based on the use of Chebyshev inequalities instead of restrictions on statistical moments. An algorithm for constructing the distribution function of the determining parameter is described. A comparison is given of the results of constructing distribution laws using the developed method and using the classical variational calculus.
Keywords: Chebyshev inequalities, complex technical system, design, determining parameter, characterization of distribution law